Nicholas Pippenger

Sex, mixability, and modularity

The assumption that different genetic elements can make separate contributions to the same quantitative trait was originally made in order to reconcile biometry and Mendelism and ever since has been used in population genetics, specifically for the trait of fitness. Here we show that sex is responsible for the existence of separate genetic effects on fitness and, more generally, for the existence of a hierarchy of genetic evolutionary modules. Using the tools developed in the process, we also demonstrate that in terms of their fitness effects, separation and fusion of genes are associated with the increase and decrease of the recombination rate between them, respectively. Implications for sex and evolution theory are discussed.

Fault tolerance in cellular automata at high fault rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate 1/2-@x (with @x>0) with degree O((1/@x^2)log(1/@x)), even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of @W(1/@x^2), even with only purely probabilistic transient faults.

Carry propagation in multiplication by constants

Suppose that a random n-bit number V is multiplied by an odd constant M ≥ 3, by adding shifted versions of the number V corresponding to the 1s in the binary representation of the constant M. Suppose further that the additions are performed by carry-save adders until the number of summands is reduced to two, at which time the final addition is performed by a carry-propagate adder. We show that in this situation the distribution of the length of the longest carry-propagation chain in the final addition is the same (up to terms tending to 0 as n → ∞) as when two independent n-bit numbers are added, and in particular the mean and variance are the same (again up to terms tending to 0). This result applies to all possible orders of performing the carry-save additions.

Stochastic service systems, random interval graphs and search algorithms

We consider several stochastic service systems, and study the asymptotic behavior of the moments of various quantities that have application to models for random interval graphs and algorithms for searching for an idle server or for an vacant or occupied waiting station. In some cases the moments turn out to involve Lambert series for the generating functions for the sums of powers of divisors of positive integers. For these cases we are able to obtain complete asymptotic expansions for the moments of the quantities in question.

M.C. escher wrap artist: aesthetic coloring of ribbon patterns

At the heart of the ideas of the work of Dutch graphic artist M.C. Escher is the idea of automation; we consider a problem that was inspired by some of his earlier and lesser known work. Specifically, a motif fragment is a connected region contained in a closed unit square. Consider a union of motif fragments and call the result an Escher tile T. One can then construct a pattern in the Euclidean plane, as Escher did, with the set of horizontal and vertical unit length translations of T. The resulting pattern gives rise to infinitely many sets of motif fragments (each set may be finite or infinite) that are related visually by way of the interconnections across boundaries of the unit squares that underly the construction; a set of related motif fragments sometimes gives the appearance of a ribbon and thus the resulting pattern in the plane is called a ribbon pattern. Eschers designs gave rise to beautiful artwork and inspired equally aesthetic combinatorial questions as well. In his sketchbooks, Escher colored the ribbon patterns with pleasing results. Coloring the ribbon patterns led naturally to a question of periodicity: is there a prototile that generates a well-colored pattern? The current work answers the question in the affirmative by way of tools from graph theory, algorithms, and number theory. We end with a list of open questions.

Local versus Global Search in Channel Graphs

Previous studies of search in channel graphs have assumed global search, for which the status of any link can be probed by the search algorithm at any time. We consider for the first time local search, for which only links to which an idle path from the source has already been established may be probed. We show that some well-known channel graphs may require exponentially more probes, on average, when search must be local than when it may be global.